Lyapunov Stability of Planar Waves to the Reaction-Diffusion Equation with a Non-Lipschitzian Reaction Term

Extending (Drábek and Takáč 2017), we investigate the Lyapunov stability of planar waves for the reaction-diffusion equation on ℝn, n ≥ 2, with a α-H€older continuous (0 < α < 1), but not necessarily smooth reaction term. We first consider an initial value problem for the equation and then construct sub- and supersolutions to the problem by a subtle modification of the planar wave. Our main result states that a bounded classical solution to the problem stays near the planar wave for all time whenever an initial data is close enough to the planar wave.